The expected number of critical percolation clusters intersecting a line segment

We study critical percolation on a regular planar lattice. Let EG(n) be the expected number of open clusters intersecting or hitting the line segment [0, n]. (For the subscript G we either take ℍ(when we restrict to the upper halfplane, or ℂ, when we consider the full lattice). Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that Eℍ((n) =An +√3/4π log(n) + o(log(n)), where A is some constant. Recently Kovács, Iglói and Cardy derived in [5] heuristically (as a special case of a more general formula) that a similar result holds for Eℂ(n) with the constant √3/4π replaced by 5√3/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of Eℍ(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of Eℂ(n).

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